Answer
Please see below.
Work Step by Step
a) We need to compare $\cosh (-x)$ with $\cosh x$:$$\cosh (-x)= \frac{e^{-x}+e^{-(-x)}}{2}=\frac{e^x+e^{-x}}{2}=\cosh x$$ Thus, $\cosh x$ is an even function.
b) We need to compare $\sinh (-x)$ with $\sinh x$:$$\sinh (-x)= \frac{e^{-x}-e^{-(-x)}}{2}=-\frac{e^x-e^{-x}}{2}=-\sinh x$$ Thus, $\sinh x$ is an odd function.
c)$$(\cosh x)^2-(\sinh x)^2=\left ( \frac{e^x+e^{-x}}{2} \right )^2 - \left ( \frac{e^x-e^{-x}}{2} \right )^2=\frac{e^{2x}+e^{-2x}+2}{4} - \frac{e^{2x}+e^{-2x}-2}{4}=\frac{4}{4}=1$$(Please note that $e^{x}e^{-x}=e^{x+ (-x)}=e^0=1$).
